Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.


(x sin y/x - y cos y/x)dx + (x cos y/x) dy = 0

Verified step by step guidance

1. Identify the given differential equation: \(\left(x \sin \frac{y}{x} - y \cos \frac{y}{x}\right) dx + \left(x \cos \frac{y}{x}\right) dy = 0\).
2. Rewrite the equation in the form \(M(x,y) dx + N(x,y) dy = 0\), where \(M(x,y) = x \sin \frac{y}{x} - y \cos \frac{y}{x}\) and \(N(x,y) = x \cos \frac{y}{x}\).
3. Check if the equation is homogeneous by verifying if \(M(tx, ty) = t^n M(x,y)\) and \(N(tx, ty) = t^n N(x,y)\) for some degree \(n\). Since the functions depend on \(\frac{y}{x}\), this suggests homogeneity of degree 1.
4. Use the substitution \(v = \frac{y}{x}\), which implies \(y = vx\) and \(dy = v dx + x dv\) to transform the equation into a separable form in terms of \(v\) and \(x\).
5. Substitute \(y\) and \(dy\) into the original equation, simplify, and then separate variables to prepare for integration.

Key Concepts

Homogeneous Differential Equations

A differential equation is homogeneous if it can be expressed so that each term is a function of the ratio y/x. This allows substitution like v = y/x to simplify the equation into a separable form, making it easier to solve.

Substitution Method (v = y/x)

By substituting v = y/x, the original variables are transformed, reducing the equation to one involving v and x only. This substitution leverages the homogeneity property and helps convert the differential equation into a separable or integrable form.

Separable Differential Equations

Once the substitution is made, the resulting equation often becomes separable, meaning it can be written as a product of a function of v and a function of x. This allows integration on both sides separately to find the solution.